Russian President Vladimir Putin says he held an in-depth discussion with U.S. President Donald Trump about Moscow's alleged meddling in the 2016 U.S. election, and that his counterpart seemed to agree with Russia's position that there were "no grounds" for such accusations.

Putin made the comments at a July 8 news conference following the Group of 20 summit in Hamburg, Germany, where he met with Trump the previous day.

The Russian president expressed cautious optimism about the prospect of improving ties with Washington that remain battered over the conflicts in Ukraine, Syria, and what U.S. intelligence agencies call a Kremlin-directed hacking and influence campaign aimed at helping elect Trump.

"There is every reason to believe that we will be able to at least partially reestablish the level of cooperation that we need," Putin told reporters.

Putin's account of his talks about alleged Russian election meddling, however, differed from the U.S side's portrayal.

Putin, who has repeatedly and emphatically denied the meddling charge, told reporters in Hamburg that Trump asked "several" questions about the alleged Russian influence campaign.

"Our position is very clear, and I stated it: there are no grounds to believe that Russia interfered in the U.S. electoral process," Putin said, adding that Trump took his position "into account" and seemed to "agree."

"But frankly, you should ask him about his opinion on this," Putin added.

U.S. Secretary of State Rex Tillerson gave a differing account, saying the issue was "something that may be an intractable disagreement at this point."

He added that Trump pressed Putin "on more than one occasion" about Moscow's involvement in the U.S. election, and that Putin had denied any interference by the Kremlin in the election.

The White House later issued a statement refuting Lavrov's claim.

Trump, who has advocated for better ties with Moscow, has said that while Russia was likely responsible for the hacking, other actors could have been behind it as well.

'Tremendous Meeting'

Originally scheduled to last 35 minutes, the talks between Putin and Trump went on for two hours and 15 minutes as they discussed the wars in Syria and Ukraine, the crisis over North Korea's nuclear and ballistic missile programs, cybersecurity, and Russia's alleged election meddling.

In his first public statement about his talks with Putin, Trump early on July 8 said that it had been a "tremendous meeting."

Putin, meanwhile, told reporters in Hamburg later in the day that "the Trump that you see on TV is very different than the real Trump."

Both sides also said Trump and Putin had focused heavily on a newly announced cease-fire deal for southwestern Syria reached by Russia and the United States on July 7.

Tillerson said that details about the cease-fire deal still need to be resolved.

But Lavrov told reporters that Russian military police would monitor the cease-fire, that it would go into effect at noon on July 9, and that a monitoring center would be set up in Jordan – which is also party to the deal.

The agreement is separate from the deal reached by Russia, Turkey, and Iran on so-called "de-escalation zones" that is being worked out in an attempt to separate combatants in Syria's six-year civil war.

Putin said in Hamburg that he believed Washington has become "more pragmatic" about the Syrian civil war under Trump, who has previously said he wants to team up with Moscow to battle Islamic State (IS) militants.

"It seems to me that the U.S. position [on Syria] has become more pragmatic. There is a comprehension that if we combine our efforts, we can achieve a lot," Putin said.

While Trump has repeatedly called for improving ties with Moscow, his administration has continued to publicly maintain pressure on Russia -- including with sanctions -- over its seizure of Ukraine's Crimean Peninsula and backing of armed separatists in eastern Ukraine.

Those punitive measures were introduced under Trump's predecessor, Barack Obama, who had a visibly icy rapport with Putin and whose administration was loathed by the Kremlin.

Trump's administration has also been dogged by the U.S. intelligence conclusion that Russia meddled in the U.S. election.

Both the U.S. Congress and the FBI are investigating the hacking and contacts between associates of the U.S. president and Russian officials, and any serious push for detente with Moscow is would almost certainly face stiff resistance from both Democratic and Republican lawmakers in Washington.

The host of the G20 summit, German Chancellor Angela Merkel, said after the conclusion of the final day that she hoped for more dialogue between Washington and Moscow.

"It is a start," Merkel told reporters, adding that there are some global problems that require cooperation between the two former Cold War foes.

"It can only be a good thing if there is an honest, frank dialogue," she said.